Local markers for crystalline topology

Abstract

Over the last few years, crystalline topology has been used in photonic crystals to realize edge- and corner-localized states that enhance light-matter interactions for potential device applications. However, the band-theoretic approaches currently used to classify bulk topological crystalline phases cannot predict the existence, localization, or spectral isolation of any resulting boundary-localized modes. While interfaces between materials in different crystalline phases must have topological states at some energy, these states need not appear within the band gap, and thus may not be useful for applications. Here, we derive a class of local markers for identifying material topology due to crystalline symmetries, as well as a corresponding measure of topological protection. As our real-space-based approach is inherently local, it immediately reveals the existence and robustness of topological boundary-localized states, yielding a predictive framework for designing topological crystalline heterostructures. Beyond enabling the optimization of device geometries, we anticipate that our framework will also provide a route forward to deriving local markers for other classes of topology that are reliant upon spatial symmetries.

8 pages, 2 figures

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• Theory of local ${\mathbb{Z}}_2$ topological markers for finite and periodic two-dimensional systems
• Versatile Control of Nonlinear Topological States in Non-Hermitian Systems
• Topological characterization of modified Kane-Mele-Rashba models via local spin Chern marker
• Scattering theory of higher order topological phases
• Family of Aperiodic Tilings with Tunable Quantum Geometric Tensor
• Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators
• Regularized universal topological markers for Dirac systems
• Topological photonic crystal fibre
• Quantitative measure of topological protection in Floquet systems through the spectral localizer
• Magnetic doping-induced second-order and first-order topological phase transition inthe photonic alloy
• Semiconductor Meta-Graphene and Valleytronics
• Dimensional crossover of class D real-space topological invariants